Min, Max and the Five-Number Summary Homework

Solution H1
For Brand C, there are 28 data entries. Here is the Five-Number Summary:

Min = 25
Q1 = 26
Med = 28
Q3 = 29
Max = 32

Here is the box plot:

The box plot shows that the center 50% of the data lies between 26 and 29, with 25% of the data falling between 28 and 29. Compared to other brands, Brand C has less variation, although there are a few boxes that have 30 or more raisins.

For Brand D, there are 36 data entries. Here is the Five-Number Summary:

Min = 23
Q1 = 27
Med = 29
Q3 = 33
Max = 38

Here is the box plot:

The box plot shows that the center 50% of the data lies between 27 and 33, with 25% of the data falling between 29 and 33. Compared to other brands, Brand D has a large number of boxes with 30 or more raisins, but it also has far greater variation than Brand C, with boxes of as few as 23 raisins.

Solution H2
Answers will vary. The comparison of four box plots generally suggests that Brands A and D offer the greatest chance for a lot of raisins in a box, although Brand C offers the most consistency and the highest minimum number of raisins in a box. The answer to this question really depends on the goals of the individual purchasing the raisins!

Solution H3
The first step is to order the lists from lowest to highest. Please note: Although each list is ordered from lowest to highest, the height and arm span measurements at the same position in the ordered list do not correspond to the same individual. (For example, the lowest height and lowest arm span are not necessarily from the same person.) Females are marked in bold.

Here is the Five-Number Summary for the 24 heights:
Min = 155
Q1 = 164
Med = 176
Q3 = 184.5
Max = 193

Here is the box plot:

b. Here is the Five-Number Summary for the 24 arm spans:

Min = 156
Q1 = 161.5
Med = 175
Q3 = 188
Max = 200

Here is the box plot:

c. Here are the Five-Number Summaries:

Males’ Heights
Min = 173
Q1 = 179
Med = 183
Q3 = 186.5
Max = 193

Females’ Heights
Min = 155
Q1 = 161
Med = 164
Q3 = 170
Max = 188

Here are the box plots:

Comparing the box plots clearly shows that the males’ heights are significantly greater than the females’. In particular, the third quartile value of females’ heights was shorter than the minimum of males’ heights, which shows that, in this survey, at least 75% of the females were shorter than the shortest male. However, the maximum height of a female is fairly close to the maximum height of a male, primarily because there was one very tall female!

d. Here are the Five-Number Summaries:

Males’ Arm Spans
Min = 173
Q1 = 177.5
Med = 188
Q3 = 191
Max = 200

Females’ Arm Spans
Min = 156
Q1 = 159.5
Med = 161.5
Q3 = 170
Max = 188

Here are the box plots:

Solution H4
The Four-Number Summary comprises the maximum, the minimum, and two values T1 and T2, which mark the endpoints of the first and second thirds of your data. The locations of T1 and T2 are determined by n, the number of values in the data set, and are also based on whether n is a multiple of three, one more than a multiple of three, or two more than a multiple of three.

If n is a multiple of three, then the position of T1 is (n / 3 + 1/2) and the position of T2 is (2n / 3 + 1/2). For example, if n = 12, T1 will be between the fourth and fifth data value (i.e., position [4.5]), and T2 will be between the eighth and ninth data value (i.e., position [8.5]). In this example, each of the three groups contains exactly four values.

If n is one more than a multiple of three, then the position of T1 is (n + 2) / 3, and the position of T2 is (2n + 1) / 3. For example, if n = 13, T1 will be the fifth position and T2 will be the ninth position. In this example, each group contains five values, if you include the endpoints.

If n is two more than a multiple of three, then the position of T1 is (n + 1) / 3, and the position of T2 is (2n + 2) / 3. For example, if n = 14, T1 will be the fifth position and T2 will be the 10th position. In this example, each group contains four values, if you don’t include the endpoints.

Other answers are also possible, depending on whether you include the endpoints of the intervals.